As I am not a professional mathematician/statistician maybe someone of you can help me writing
the correct mathematical notation for following procedure/task:

Assuming the probability function like the standard normal distribution with
is

The cumulative density functions for x are as follows:

Then

So far as I remember to take the difference between two CDFs of a normal distribution works only if both x values are positive (if ). Is that correct? If yes
how can this criterion be corretly implemented in the equations above (correct notation)?
What's about the equations? Are they correct from a mathematical point of view and is the notation correct in general? How can it be improved?